Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Stabilization of nonlinear systems with unknown delays via delay-adaptive neural operator approximate predictors

Luke Bhan, Miroslav Krstic, Yuanyuan Shi

TL;DR

This paper addresses stabilization of nonlinear systems with unknown constant actuator delays by using predictor-feedback implemented via neural-operator approximators. It provides rigorous stability guarantees: semi-global practical stability when the actuator input is measurable and local practical stability when it is unmeasured, with bounds depending on the predictor error $ε$ and delay-estimation error; these results leverage a PDE-ODE cascade, backstepping, and Lyapunov–Krasovskii analysis. Neural operators (e.g., DeepONet, Fourier Neural Operator) are shown to approximate the predictor map with arbitrarily small $ε$, enabling the theoretical guarantees to be realized in practice. Numerical experiments on a biological protein clock and a Chemostat demonstrate stability under approximate predictors, good generalization of the neural operators, and speedups up to $15\times$ over traditional solvers, highlighting substantial practical impact for real-time delay compensation in nonlinear systems.

Abstract

This work establishes the first rigorous stability guarantees for approximate predictors in delay-adaptive control of nonlinear systems, addressing a key challenge in practical implementations where exact predictors are unavailable. We analyze two scenarios: (i) when the actuated input is directly measurable, and (ii) when it is estimated online. For the measurable input case, we prove semi-global practical asymptotic stability with an explicit bound proportional to the approximation error $ε$. For the unmeasured input case, we demonstrate local practical asymptotic stability, with the region of attraction explicitly dependent on both the initial delay estimate and the predictor approximation error. To bridge theory and practice, we show that neural operators-a flexible class of neural network-based approximators-can achieve arbitrarily small approximation errors, thus satisfying the conditions of our stability theorems. Numerical experiments on two nonlinear benchmark systems-a biological protein activator/repressor model and a micro-organism growth Chemostat model-validate our theoretical results. In particular, our numerical simulations confirm stability under approximate predictors, highlight the strong generalization capabilities of neural operators, and demonstrate a substantial computational speedup of up to 15x compared to a baseline fixed-point method.

Stabilization of nonlinear systems with unknown delays via delay-adaptive neural operator approximate predictors

TL;DR

This paper addresses stabilization of nonlinear systems with unknown constant actuator delays by using predictor-feedback implemented via neural-operator approximators. It provides rigorous stability guarantees: semi-global practical stability when the actuator input is measurable and local practical stability when it is unmeasured, with bounds depending on the predictor error and delay-estimation error; these results leverage a PDE-ODE cascade, backstepping, and Lyapunov–Krasovskii analysis. Neural operators (e.g., DeepONet, Fourier Neural Operator) are shown to approximate the predictor map with arbitrarily small , enabling the theoretical guarantees to be realized in practice. Numerical experiments on a biological protein clock and a Chemostat demonstrate stability under approximate predictors, good generalization of the neural operators, and speedups up to over traditional solvers, highlighting substantial practical impact for real-time delay compensation in nonlinear systems.

Abstract

This work establishes the first rigorous stability guarantees for approximate predictors in delay-adaptive control of nonlinear systems, addressing a key challenge in practical implementations where exact predictors are unavailable. We analyze two scenarios: (i) when the actuated input is directly measurable, and (ii) when it is estimated online. For the measurable input case, we prove semi-global practical asymptotic stability with an explicit bound proportional to the approximation error . For the unmeasured input case, we demonstrate local practical asymptotic stability, with the region of attraction explicitly dependent on both the initial delay estimate and the predictor approximation error. To bridge theory and practice, we show that neural operators-a flexible class of neural network-based approximators-can achieve arbitrarily small approximation errors, thus satisfying the conditions of our stability theorems. Numerical experiments on two nonlinear benchmark systems-a biological protein activator/repressor model and a micro-organism growth Chemostat model-validate our theoretical results. In particular, our numerical simulations confirm stability under approximate predictors, highlight the strong generalization capabilities of neural operators, and demonstrate a substantial computational speedup of up to 15x compared to a baseline fixed-point method.

Paper Structure

This paper contains 17 sections, 16 theorems, 136 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Predictor feedback designs and implementations
  3. Operator learning in control
  4. Notation
  5. Technical background
  6. Delay-adaptive control
  7. Neural operators
  8. Neural operator approximate predictors
  9. Stability analysis under approximate predictors
  10. Numerical Experiments
  11. Control in presence of an unmeasured distributed input
  12. Numerical Example: Biological Chemostat
  13. Conclusion
  14. Reverse Young's Inequality
  15. Proof of Lemma \ref{['lemma:backstepping-lemma-distributed-input']}
  16. ...and 2 more sections

Key Result

Theorem 1

lanthaler2024nonlocalitynonlinearityimpliesuniversality Let $\Omega_u \subset \mathbb{R}^{d_{u_1}}$ and $\Omega_v \subset \mathbb{R}^{d_{v_1}}$ be two bounded domains with Lipschitz boundary. Let $\Psi: C^0(\overline{\Omega_u};\mathbb{R}^{d_{u_1}}) \rightarrow C^0(\overline{\Omega_v}; \mathbb{R}^{d_ for all values $y \in \Omega_v$.

Figures (2)

  • Figure 1: Simulation of the plant \ref{['eq:main-problem']} with various approximate predictors. The initial state is $X(0) = [0.03, 30]$, delay is $D=1$, $\hat{D}(0) = 2$, $\gamma=1000$, $b=1$. The black line indicates the numerical predictor, the blue-line is the predictor without delay-adaptation, the red line indicates the DeepONet predictor with higher error and the green line indicates the DeepONet predictor trained to optimality.
  • Figure 2: Chemostat \ref{['eq:chemostat']} with $Z(0)=2,S(0)=2$, $\hat{D}(0)=1.8$, with target setpoint at the unstable equilibrium $Z^\ast=3,S^\ast=2$karafyllis2025globalstabilizationchemostatsnonzero. For true delay $D=1.6$ s, comparison of (i) numerical predictor (black), (ii) open-loop without adaptation, unstable (blue), and (iii) FNO predictor, stable (green).

Theorems & Definitions (18)

  • Definition 1
  • Theorem 1
  • Definition 2
  • Lemma 1
  • Theorem 2
  • Theorem 3
  • Lemma 2
  • Theorem 4
  • Lemma 3
  • Lemma 4
  • ...and 8 more